Let $A$ be an abelian variety over a number field $K$ and let $\mathcal{A}$ denote its Neron model over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field, $\mathcal{A}_v = \mathcal{A} \times _{\mathcal{O}_K} k_v$ the special fiber of the reduction of $A$ at $v$, and $\mathcal{A}_v^0$ the connected component of the identity section in the special fiber $\mathcal{A}_v$. Then it is well known that there is a finite group-scheme $\Phi _{A,v}$, s.t. we have an exact sequence of $k_v$-group-schemes

$1 \rightarrow \mathcal{A}_v^0 \rightarrow \mathcal{A}_v \rightarrow \Phi _{A,v} \rightarrow 1$.

Is it true that

$\Phi _{A,v}(k_v) = \mathcal{A}_v(k_v) / \mathcal{A}_v^0(k_v)$?

Remark:

The motivation for my question is the definition of the Tamagawa number $c_{A,v}$, which in one source is defined as the cardinality of $\Phi _{A,v}(k_v)$ and in another source as the cardinality of the quotient group $\mathcal{A}_v(k_v) / \mathcal{A}_v^0(k_v)$. And a priori one only knows that $\Phi _{A,v}(k_v) = H^0(\mathcal{A}_v(\bar k_v) / \mathcal{A}_v^0(\bar k_v))$.